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Identical expressions
xdx/(x+ one)(x+ three)(x+ five)
xdx divide by (x plus 1)(x plus 3)(x plus 5)
xdx divide by (x plus one)(x plus three)(x plus five)
xdx/x+1x+3x+5
xdx divide by (x+1)(x+3)(x+5)
Similar expressions
xdx/(x+1)(x+3)(x-5)
xdx/(x+1)(x-3)(x+5)
xdx/(x-1)(x+3)(x+5)

Solving integrals/ xdx/(x+1)(x+3)(x+5)

Integral of xdx/(x+1)(x+3)(x+5) dx

The solution

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  1                         
  /                         
 |                          
 |    x                     
 |  -----*(x + 3)*(x + 5) dx
 |  x + 1                   
 |                          
/                           
0

$$\int\limits_{0}^{1} \frac{x}{x + 1} \left(x + 3\right) \left(x + 5\right)\, dx$$

Integral(((x/(x + 1))*(x + 3))*(x + 5), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Rewrite the integrand:
  
  $\frac{x}{x + 1} \left(x + 3\right) \left(x + 5\right) = x^{2} + 7 x + 8 - \frac{8}{x + 1}$
2. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 7 x\, dx = 7 \int x\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $\frac{7 x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 8\, dx = 8 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{8}{x + 1}\right)\, dx = - 8 \int \frac{1}{x + 1}\, dx$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
    So, the result is: $- 8 \log{\left(x + 1 \right)}$
  The result is: $\frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 8 x - 8 \log{\left(x + 1 \right)}$
Method #2
1. Rewrite the integrand:
  
  $\frac{x}{x + 1} \left(x + 3\right) \left(x + 5\right) = \frac{x^{3} + 8 x^{2} + 15 x}{x + 1}$
2. Rewrite the integrand:
  
  $\frac{x^{3} + 8 x^{2} + 15 x}{x + 1} = x^{2} + 7 x + 8 - \frac{8}{x + 1}$
3. Integrate term-by-term:
  1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int x^{2}\, dx = \frac{x^{3}}{3}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int 7 x\, dx = 7 \int x\, dx$
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
    So, the result is: $\frac{7 x^{2}}{2}$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 8\, dx = 8 x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{8}{x + 1}\right)\, dx = - 8 \int \frac{1}{x + 1}\, dx$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
    So, the result is: $- 8 \log{\left(x + 1 \right)}$
  The result is: $\frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 8 x - 8 \log{\left(x + 1 \right)}$
Method #3
1. Rewrite the integrand:
  
  $\frac{x}{x + 1} \left(x + 3\right) \left(x + 5\right) = \frac{x^{3}}{x + 1} + \frac{8 x^{2}}{x + 1} + \frac{15 x}{x + 1}$
2. Integrate term-by-term:
  1. Rewrite the integrand:
    
    $\frac{x^{3}}{x + 1} = x^{2} - x + 1 - \frac{1}{x + 1}$
  2. Integrate term-by-term:
    1. The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x^{2}\, dx = \frac{x^{3}}{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- x\right)\, dx = - \int x\, dx$
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      So, the result is: $- \frac{x^{2}}{2}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + 1}\right)\, dx = - \int \frac{1}{x + 1}\, dx$
      
      Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
      
      So, the result is: $- \log{\left(x + 1 \right)}$
    The result is: $\frac{x^{3}}{3} - \frac{x^{2}}{2} + x - \log{\left(x + 1 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{8 x^{2}}{x + 1}\, dx = 8 \int \frac{x^{2}}{x + 1}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x^{2}}{x + 1} = x - 1 + \frac{1}{x + 1}$
    2. Integrate term-by-term:
      
      The integral of $x^{n}$ is $\frac{x^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int x\, dx = \frac{x^{2}}{2}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int \left(-1\right)\, dx = - x$
      
      Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
      
      The result is: $\frac{x^{2}}{2} - x + \log{\left(x + 1 \right)}$
    So, the result is: $4 x^{2} - 8 x + 8 \log{\left(x + 1 \right)}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{15 x}{x + 1}\, dx = 15 \int \frac{x}{x + 1}\, dx$
    1. Rewrite the integrand:
      
      $\frac{x}{x + 1} = 1 - \frac{1}{x + 1}$
    2. Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, dx = x$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{1}{x + 1}\right)\, dx = - \int \frac{1}{x + 1}\, dx$
      
      Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{u}\, du$
      
      The integral of $\frac{1}{u}$ is $\log{\left(u \right)}$ .
      
      Now substitute $u$ back in:
      
      $\log{\left(x + 1 \right)}$
      
      So, the result is: $- \log{\left(x + 1 \right)}$
      
      The result is: $x - \log{\left(x + 1 \right)}$
    So, the result is: $15 x - 15 \log{\left(x + 1 \right)}$
  The result is: $\frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 8 x - 8 \log{\left(x + 1 \right)}$
Add the constant of integration:

$\frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 8 x - 8 \log{\left(x + 1 \right)}+ \mathrm{constant}$

The answer is:

$\frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 8 x - 8 \log{\left(x + 1 \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                             
 |                                                      3      2
 |   x                                                 x    7*x 
 | -----*(x + 3)*(x + 5) dx = C - 8*log(1 + x) + 8*x + -- + ----
 | x + 1                                               3     2  
 |                                                              
/

$$\int \frac{x}{x + 1} \left(x + 3\right) \left(x + 5\right)\, dx = C + \frac{x^{3}}{3} + \frac{7 x^{2}}{2} + 8 x - 8 \log{\left(x + 1 \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

71/6 - 8*log(2)

$$\frac{71}{6} - 8 \log{\left(2 \right)}$$

71/6 - 8*log(2)

$$\frac{71}{6} - 8 \log{\left(2 \right)}$$

71/6 - 8*log(2)

Numerical answer [src]

6.28815588885377

6.28815588885377

The graph

Use the examples entering the upper and lower limits of integration.

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Identical expressions

Similar expressions

Integral of xdx/(x+1)(x+3)(x+5) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #3